Classification, Koszulity and Artin-Schelter Regularity of certain Graded Twisted Tensor Products

Let $\mathbb{k}$ be an algebraically closed field. We classify all of the quadratic twisted tensor products $A \otimes_τ B$ in the cases where $(A, B) = (\mathbb{k}[x], \mathbb{k}[y])$ and $(A, B) = (\mathbb{k}[x, y], \mathbb{k}[z])$. We determine when a quadratic twisted tensor product of this form is Koszul, and when it is Artin-Schelter regular.

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